Study of the Stability Properties for a General Shape of Damped Euler – Bernoulli Beams under Linear Boundary Conditions

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Introduction
The stabilization of damped beams submitted to vibrations has been one of the main research topics in smart materials and structures.A beam has two spatially nonhomogeneous damping terms.The first is often called structural damping, while the second acts opposite to the velocity and is called viscous damping.When these damping coefficients are constant, it is known that viscous damping causes a constant attenuation rate for all frequencies of vibrations.In this paper, we study a general shape of Euler-Bernoulli beams with variable coefficients under the influence of a viscous damping.This beam is clamped at one end and controlled at its free end in force by a linear combination of the velocity, rotation, and velocity of rotation and in a moment by a linear combination of the velocity, velocity of rotation, and the position term.Let y x; ð tÞ the transversal deviation at position x and time t.The equations are defined by the following: ð Þ is the mass density of the beam and the function EI : ð Þ is its flexural rigidity satisfying m : ð Þ >0 and EI : ð Þ>0.Throughout this paper, we assume that Moreover, γ : ð Þ is a continuous coefficient function of feedback damping that is assumed to satisfy the condition Our interest in this work is to generalize the works carried out by several authors on damped Euler-Bernoulli beams.
Aouragh and Yebari [1] studied the system (1) without damping (γ ¼ 0) and with the condition ζ 12 >0.In their article, the authors showed that there is a sequence of generalized eigenfunctions that form a Riesz basis of the appropriate Hilbert space and that there is exponential stability under certain conditions relating to the feedback coefficients.
For our part, we study the system (1) with the presence of a viscous damping γ x ð Þ>0 and the condition ζ 12 ≥ 0. To do this, we analyze two situations as follows: Case 1: We consider the case where ζ 12 ¼ 0, which implies with the condition (2) that ζ 11 ¼ ζ 22 ¼ 0. Our investigation of this case allows us to deduce the results of asymptotic expressions by Guo [2], Koffi et al. [3], Touré et al. [4], and Wang et al. [5].In these cases, the authors have concluded for the exponential stability.Case 2: We study the case where ζ 12 >0 and Here, first, we find a general asymptotic expression of the eigenvalues of the system where we could deduce those found by Jean-Marc et al. [6] and Aouragh and Yebari [1] in the constant case.Next, by proceeding by Wang [7], the exponential stability of the system (1) is obtained when γ is non-negative in [0,1], and we discuss the case that γ is continuous and indefinite on the interval [0, 1].
In the second case, we consider non-zero all the coefficients in the boundary conditions, which made the calculations very complex.The calculations have been carried out meticulously to obtain the expected result.
During our analysis, we used the asymptotic method of Wang, which is essential for our study.This method comes from the work of Birkhoff [8,9].We also find in the literature several authors who have used this approach to study Euler-Bernoulli beam equations with variable coefficients (see Wang et al. [5], Wang [7], Guo [10], Guo and Wang [11]).
The contents of the paper are arranged as follows: in Section 2, we formulate the system (1) into an abstract Cauchy problem in Hilbert state space and discuss some basic properties of the system.In Section 3, we study the spectrum properties for each case.The Riesz basis property and the exponential stability in the general case are concluded in Section 4.

Semigroup Formulation
In this part, we study the well-posedness of the system (1) and deduce some properties of the operator of this system.We introduce this Hilbert spaces as follows: with the inner product Here, : k k H denotes the corresponding norm.The spaces L 2 0; ð 1Þ and H k 0; ð 1Þ are defined as follows: Next, we define an unbounded linear operator A : with the domain The system (1) can be formally written as a first-order evolution problem 2 Abstract and Applied Analysis Furthermore, we introduce the linear operator B defined on D A ð Þ by the following: Notice that A 0 denotes the operator where γ x ð Þ ¼ 0. In the study of Aouragh and Yebari [1], it is shown that the linear operator A 0 generates a C 0 -semigroup of contractions on H.Moreover, the linear operator B is bounded on H.In fact, where Now, we deduce the well-posedness of the system through the following theorem.
Theorem 2.1.The operator A defined by Equations ( 9) and (10) generates a C 0 -semigroup of contraction e At f g t≥0 on H and has compact resolvent and 0 2 ρ A ð Þ: Therefore, the spectrum σ A ð Þ consists entirely of isolated eigenvalues.
Proof.Let U ¼ y; y t ð Þ T 2 DðA).Integrating twice by parts and taking real part, we obtain the following: Thus, A is a dissipative operator.Moreover, the linear operator B is bounded on H and A 0 generates a C 0 -semigroup of contractions.Then, according to perturbation theory by a bounded linear operator (see Pazy [12] P.76), the operator Now, we show that the operator A −1 exists.For any G ¼ u; ð vÞ 2 H, we must find a unique This leads us to the following system: A direct calculation shows that where f 1 ð Þ and f x 1 ð Þ are defined as follows: Abstract and Applied Analysis Obviously, we have F ¼ f ; ð gÞ 2 D A ð Þ.Therefore, we get the following: Thus, A −1 exists and consequently we have 0 2 ρ A ð Þ and Sobolev's embedding theorem implies that A −1 is a compact operator on H.

General Spectral Analysis
and Our work shall follow the results from Wang [5].Let λ 2 σ A ð Þ be an eigenvalue of the operator A and let where the prime refers to the derivative with respect to space variables.This equation leads to the following system: First, we use a space transformation to transform the coefficient function appearing with ϕ in the first expression of Equation (23) into a constant (see Guo [10]).
Abstract and Applied Analysis A direct computation gives this system as follows: where Next, we make an invertible state transformation by Naimark [13].
The function g satisfies the following: By setting λ ¼ ρ 2 =h 2 , the first equation of the system (29) becomes the following: Now, to solve the eigenvalue problem (29), we divide the complex plane into eight distinct sectors by Naimark [13].
Let ω 1 ; ω 2 ; ω 3 , and ω 4 be the roots of characteristic equation In sector S 1 , we name the roots as follows: that satisfy the inequalities (34).Similarly, the choices can also be made for other sectors.In the following, we study the asymptotic behavior of the eigenvalues specific to sectors S 1 and S 2 because this will be verified in the other Abstract and Applied Analysis sectors with similar arguments.For this, we will use the following lemma: Lemma 3.1.For ρ 2 S k , with |ρj large enough, the equation has four linearly independent asymptotic fundamental solutions and hence their derivatives for s ¼ 1; 2; 3; 4 and j ¼ 1; 2; 3 are given by the following: where for s ¼ 1; 2; 3; 4 with Proof.The reader can refer to the study of Touré et al. [4].□ For convenience, we introduce the notation a =2, then we have the following inequalities: Re Furthermore, substituting Equations ( 37) and (38) into the boundary conditions (29), we obtain asymptotic expressions for the boundary conditions for large enough ρ j j as follows: where Notice that λ ¼ ρ 2 =h 2 ≠ 0, is the eigenvalue of Equation (29) if and only if ρ satisfies the characteristic equation as follows: By substitution, the following expression is obtained: Abstract and Applied Analysis In sector S 1 , the choices are such that Developing the determinant, and after a straightforward computation, we obtain the following: Noting that the characteristic determinant in sector S 1 becomes the following: Then, we have Θ −10 ¼ 2, Θ 10 ¼ 2 and Θ 00 ¼ 0, (see Jean-Marc et al. [6]).
Theorem 3.3.The boundary conditions of the eigenvalue problem (29) are strongly regular.Therefore, the eigenvalues are asymptotically simple and separated.Now, we study the asymptotic behavior for the eigenvalues λ n of problem (29).The equation which can also be rewritten as follows: The equation below: has the solutions given by the following: Let ρk be the solution of Equation ( 53): According to Rouché's theorem (see Naimark [13]), we obtain the following expression: where K is a sufficiently large positive integer.By substituting it in Equation (52) and using the equality e −ρ k ω 2 ¼ − e ρ k ω 2 ; we get the following: Moreover, expanding the exponential function according to its Taylor series, we get the following: Note that λ k ¼ ρ2 k =h 2 and in sector S 1 ; ω 2 ¼ e i 1 4 π and ω 2 2 ¼ i.So, we have the following: Abstract and Applied Analysis where k ¼ K; K þ 1; ⋅⋅⋅, with K large enough.Now, in sector S 2 , the eigenvalues of the problem (29) can be obtained by a similar computation with the choices.
By a similar calculation to the one done in sector S 1 , we have the following: with K a large enough integer.In the sector S 2 , using ω 2 ¼ e i 3 4 π and ω 2 2 ¼ − i, we obtain for k ¼ Remark 3.4.By referring to Naimark [13], we can say that the eigenvalues generated by the sectors S n coincide with those determined in the sectors S 1 and S 2 .
Combining with Equations ( 59) and (63), we obtain the asymptotic expression Theorem 3.5.An asymptotic expression of the eigenvalues of the problem (29) is given by the following: where k ¼ K; K þ 1; ⋅⋅⋅, with K a large enough integer, and ⋅⋅⋅Þ with sufficiently large modulus are simple and distinct except for finitely many of them, and satisfy From Equation (67), we find these results already known.
8 Abstract and Applied Analysis asymptotic expression (67) becomes the following: This case corresponds to the one studied by Wang et al. [5], where the result (69) is obtained.
Guo [2] studied this system.His work yielded a similar asymptotic expression as Equation (67) in the uniform case Example 4. For ζ 21 >0 and α; β ≥ 0, we get the following system: This case was investigated by Koffi et al. [3], where the asymptotic expression (67) was found.
Note that in the cited examples, the authors proved that the systems are exponentially stable.

Study of General Case
In this subsection, we study the system (1) with the conditions ζ 12 >0 and Following the previous approach, we determine a general asymptotic expression of the eigenvalues of the system (1) and deduce those established by Aouragh and Yebari [1] and Touré et al. [4].
Let λ 2 σ A ð Þ be an eigenvalue of the operator A of the system (1).
Abstract and Applied Analysis The system (73) is rewritten as First, by using the space transformation as follows: we have x are defined in Equations (26) and ( 27) Then, by making the invertible state transformation the system (76) can be written as follows, for any 0<z <1 : where By setting λ ¼ ρ 2 =h 2 , the first equation of the system (79) becomes the following: which has four linearly independent asymptotic fundamental solutions given by Equations ( 37) and (38).To solve the eigenvalue problem (79), we make a study in the sector S 1 which is defined like this Let ω 1 ; ω 2 ; ω 3 , and ω 4 be the roots of characteristic equation ω 4 þ 1 ¼ 0 that are arranged so that In sector S 1 , the choices are the same as in Equation ( 35).Substituting Equations ( 37) and (38) into the boundary conditions (79); we obtain asymptotic expressions for the boundary conditions for large enough ρ j j: Abstract and Applied Analysis with By substituting the expression (84) into (44), the characteristic determinant of the eigenvalue problem (79) is given by the following: 12 Abstract and Applied Analysis the determinant, and after a straightforward computation, we obtain the following: Then, we have τ 26 , and Θ 00 ¼ 0.
Theorem 3.6.The boundary conditions of the eigenvalue problem (79) are strongly regular.Therefore, the eigenvalues are asymptotically simple and separated.Now, we study the asymptotic behavior for the eigenvalues λ n of problem (79).The equation which can also be rewritten as follows: The equation e ρω 2 þ ie −ρω 2 ¼ 0; has the solutions given by the following: Let ρk be the solution of Equation (93).According to Rouché's theorem (see Naimark [13]), we obtain the following expression: where K is a sufficiently large positive integer.By substituting it in Equation (92) and using the equality e −ρ k ω 2 ¼ ie ρ k ω 2 ; we get the following: Moreover, expanding the exponential function according to its Taylor series, we get the following: Note that λ k ¼ ρ2 k =h 2 and in sector S 1 ; ω 2 ¼ e i 1 4 π and ω 2 2 ¼ i.So, we have the following: By applying the same proof to the sector S 2 , the eigenvalues of the problem (79) can be obtained by a similar computation with the same choices as in Equation (60).Hence, in the sector S 2 , the characteristic determinant Δ ρ ð Þ is as follows: By a calculation similar to the one done in sector S 1 , we have the following: Abstract and Applied Analysis In the sector S 2 , ω 2 ¼ e i 3 4 π and ω 2 2 ¼ − i, we obtain the following: where k ¼ K; K þ 1; …, with K a large enough integer, and ⋅⋅⋅Þ with sufficiently large modulus are simple and distinct except for finitely many of them, and satisfy According to the condition (2), the constant l is positive.From Equation (105), we find the following results: This system has been studied by Jean-Marc et al. [6].If and Equation (105) becomes the following: The asymptotic expression (107) is equivalent to the one in Theorem 8 by Jean-Marc et al. [6].
Example 2. Suppose that ζ 12 >0 and γ ¼ 0, the system (1) is equivalent to the following: 14 Abstract and Applied Analysis has been studied by Aouragh and Yebari [1], where they get the same result as Equation (105) in the uniform case m Remark 3.8.In the cited examples, the authors proved that the systems are exponentially stable.

Riesz Basis Property and Exponential
Stability of the System in the General Case In this section, we consider the system (1) in the general case with the conditions 4.1.Riesz Basis Property of the System.Here, we discuss the Riesz basis property of the eigenfunctions of the operator A of the system by following an idea due to Wang et al. [5].We begin by considering a bounded invertible operator defined on H by the following: Then, we define the ordinary differential operator as follows: Let H be the Hilbert space defined by Wang [7] and define the operator A in H by the following: Let η 2 σ A ð Þ be an eigenvalue of A and f ; ð gÞ the corresponding eigenfunction.Then we obtain g ¼ ηf and f satisfies as follows: Now by taking we see that ψ ¼ λϕ and ϕ satisfies Hence, we have that: Theorem 4.1.Let operator A be defined by Equations ( 9) and ( 10):Then the eigenvalues of operatorAare all simple except for finitely many of them, and the generalized eigenfunctions of operatorAform a Riesz basis for the Hilbert state spaceH.
Proof.We have shown that the boundary problem (79) is strongly regular (see Theorem 3.6).Therefore, the eigenvalues are separated and simple except for finitely many of them.Thus, the first statement follows.Moreover, Theorem 4.1.1 in the study of [7] (1).Referring to the study of Curtain and Zwart [14], we note that the Riesz basis property implies the spectrum-determined growth condition and Equation (105) describes the asymptote of σ A ð Þ, for any small ε>0 there are only finitely many eigenvalues of A in the following half-plane: Following Theorem 2.4 in the study of Gao [10], all the properties of operator A found above, allow us to claim that for the semigroup e At f g t≥0 generated by A the spectrumdetermined growth condition holds the following: By using Theorem 4.1, the exponential stability of system (1) can be concluded.
of system (1) satisfies E t ð Þ ≤ ME 0 ð Þe −wt ; 8t ≥ 0, for any initial condition y x; 0 ð Þ; y t x; Proof.The operator A is dissipative.In effect, For We also know that A generates a semigroup of contractions e At f g t≥0 on H.Moreover, the spectrum of the operator A has an asymptote as follows: Then, the study of the exponential stability is equivalent to verifying that Reλ<0.Let λ ¼ ib with b 2 R * be an eigenvalue of operator A on the imaginary axis and Ψ ¼ ϕ; ψ ð Þ T be the corresponding eigenfunction, then ψ ¼ λϕ.
Re AΨ ; Then, Moreover ψ ¼ λϕ, then from Equations ( 74) and (124); the following differential equation is satisfied by ϕ x ð Þ: We show with the help of Rolle's Theorem that the null function is the unique solution of Equation (125).For this, the reader is invited to follow a method used by Jean-Marc et al. [6].
From Theorem 4.1 and the spectrum-determined growth condition, the system is exponentially stable for any □ 16 Abstract and Applied Analysis using an idea of Wang [7], we study the situation where γ x ð Þ is continuous and indefinite in [0,1].We have the following theorem.
Hence, the operator A can be written as then the system (1) is exponentially stable.
Proof.According to Section 2 of this paper, the operator B − is bounded and symmetric, so it is a self-adjoint operator (see Brezis [15]) and By Theorem 4.2 and the definition of operator A þ , e A þ t È É is a contraction semigroup and s A þ ð Þ<0.Applying the perturbation theory of linear operator's semigroup (see Pazy [12]), we have λ 2 σ A ð Þ whenever Reλ>s A þ ð Þþ B − k k.Furthermore, Theorem 4.1 ensures that Therefore, the system (1) is exponentially stable if

Conclusion
During our analysis, after showing the basic properties of the operator, we established an asymptotic expression of the operator, which generalizes several other results.Then, using the Riesz basis property, we have shown that the system is exponentially stable in the general case following the sign of γ.In addition, our results can also be extended to other Euler-Bernoulli beam problems.